A mathematical model for the transmission of co-infection with COVID-19 and kidney disease

The world suffers from the acute respiratory syndrome COVID-19 pandemic, which will be scary if other co-existing illnesses exacerbate it. The co-occurrence of the COVID-19 virus with kidney disease has not been available in the literature. So, further research needs to be conducted to reveal the transmission dynamics of COVID-19 and kidney disease. This study aims to create mathematical models to understand how COVID-19 interacts with kidney diseases in specific populations. Therefore, the initial step was to formulate a deterministic Susceptible-Infected-Recovered (SIR) mathematical model to depict the co-infection dynamics of COVID-19 and kidney disease. A mathematical model with seven compartments has been developed using nonlinear ordinary differential equations. This model incorporates the invariant region, disease-free and endemic equilibrium, along with the positivity solution. The basic reproduction number, calculated via the next-generation matrix, allows us to assess the stability of the equilibrium. Sensitivity analysis is also utilised to understand the influence of each parameter on disease spread or containment. The results show that a surge in COVID-19 infection rates and the existence of kidney disease significantly enhances the co-infection risks. Numerical simulations further clarify the potential outcomes of treating COVID-19 alone, kidney disease alone, and co-infected cases. The study of the potential model can be utilised to maximise the benefits of simulation to minimise the global health complexity of COVID-19 and kidney disease.


Model formulation
We consider a deterministic seven-compartmental human population (Fig. 1).The total population is divided into seven sub-classes, which are susceptible population (S), infectious individuals with COVID-19 (I c ) , infected by the primary stage of the kidney (I k ), infected by end-stage kidney disease (I kd ) , co-infected with COVID-19 and primary stage of kidney disease (I kc ),co-infected with COVID-19 and end-stage kidney disease (I kdc ) , individ- uals who have recovered from COVID-19 (R) .We assume that the rate of increase in the susceptible population stems from a recruitment rate represented by , while there's a natural mortality rate µ present across all classes.In the total susceptible population, individuals can get kidney disease with a contact rate of φ 2 from a kidney disease only infected or co-infected person with the force of infection of kidney disease, f k = φ 2 [I k +θ (I kd +I kc +I kdc )] N , and join I k state variable.Similarly, individuals can get COVID-19 with a contact rate of φ 1 from a COVID-19- only infected or co-infected person with the force of infection of COVID-19 f c = φ 1 [I c +γ (I kdc +I kc )] N , and join I c state variable.Kidney disease only infected individuals can also get an additional COVID-19 infection with the force of infection f c and join the co-infected compartment (I kc ).The co-infected compartment increases because individuals that come from COVID-19 only infected compartment when kidneys infect them with f k the force of infection.In this context, θ is the parameter adjusting for the enhanced transmission of kidney disease among co-infected individuals and those in the end-stage of the disease, and γ denotes the parameter accounting for the amplified transmissibility of COVID-19 in co-infected persons.Parameters σ 1 ,σ 2 are represented progression rates to fully increased kidney disease by compartments I k and I kc , respectively.The parameters τ 1 , τ 2 and τ 3 indicate recovery rates from COVID-19 for individuals in compartments (I c ) , co-infected with COVID-19 and primary stage of kidney disease (I kc ) , co-infected with COVID-19 and end-stage kidney disease (I kdc ) respectively and α 1 ,α 2 parameters denote adjustments for the susceptibility of individuals with kidney disease to contracting a COVID-19 infection.

Analytical analysis
We studied how COVID-19 and kidney disease impact each other by examining them separately first.After understanding each individually, they're combined to see the overall effect.The goal is to ensure the combined results are accurate and logical.
COVID-19-only model: when we exclude kidney disease infections, we can formulate a COVID-19-specific sub-model from the full disease model; we get I k = 0, I kc = 0, I kd = 0, I kdc = 0 Theorem 1 All the populations of the system with positive initial conditions are nonnegative.
Assume S(0) > 0, I C (0) > 0, R(0) > 0 are positive for time t > 0 and for all nonnegative parameters.From the initial condition, all the state variables are nonnegative at the initial time; then, t > 0 To show the solutions of the model, as it is positive, first, we take dS dt from equation  An invariant region is identified to demonstrate that the solution remains within certain bounds.This invariant region provides a constraint ensuring that the solution does not exceed these limits; we have As,t → ∞ , we get 0 ≤ N ≤ � µ , the theory of differential equation 27 in the region.Z c = {(S, I c , R)ǫR 3 + : N ≤ � µ } , For the autonomous system representing the COVID-19-only model, given by (2), any solution that starts in Z c will stay within Z c for all t ≥ 0. Based on Cheng et al., this means that Z c acts as a stable and attractive region.Therefore, according to Naicker et al., the dynamics of model ( 2) are both mathematically sound and relevant to epidemiology, and it is appropriate to study its tabiliz within Z c .

Stability analysis of equilibrium states:
In the only COVID-19 sub-model, the equilibrium state is reached when the following conditions are met For the isolated COVID-19 model represented by the system (2), the state without any active disease (termed the disease-free equilibrium or DFE) is derived by setting each component of the system (2) to zero.At this DFE, neither infections nor recoveries are present.
Therefore, for the stand-alone COVID-19 model ( 2), the DFE is described � c = (S, I C , R) = ( � µ , 0, 0) The sub-model's basic reproduction number is the average number of secondary infections caused by a single COVID-19-infected person in a totally susceptible population.The system (2) calculates it using the next-generation matrix.
The basic reproduction number, R 0c , represents the average number of people one infected individual is expected to infect over their entire infectious period within a completely susceptible population.

Theorem 3
For the kidney disease sub-model, the point of equilibrium without the disease is represented as 0c , remains stable as long as the basic reproduction number R oc is less than 1. www.nature.com/scientificreports/ The Jacobian matrix is tabiliz to ascertain the equilibrium points' local stability.For sub-model ( 2), the Jacobian matrix is formulated as The Jacobian matrix for the sub-model, when evaluated at the disease-free equilibrium point 0c , is expressed as In this context, one of the eigenvalues for 0c is = −µ .The other eigenvalues can be conveniently derived from the associated submatrix.
To confirm the local stability of the disease-free equilibrium point, two conditions need to be met: (i) The trace of J 1 should be less than zero. (ii) The determinant of J 1 should be greater than zero.The trace is Trc (J 1 ) = −(τ 1 + 2µ), which is less than zero.
As a result, the COVID-19 sub-model's disease-free equilibrium point is asymptotically stable.Theorem 5.The COVID-19 submodel has an isolated endemic equilibrium point if R 0c > 1.The endemic equilibrium point of the COVID-19 sub-model is the solution of the system of equation in (4).
To solve this system of equations, we express it in terms of Now, So, using (4) The conclusion drawn is that the infection force f * c will be positive at the endemic equilibrium point 0c only when R oc > 1 .With this, we have effectively demonstrated the related theorem.

Theorem 5 Analysis of the Global Stability Analysis for the Endemic Equilibrium Point.
The endemic equilibrium point c undergoes a global stability analysis using the Lyapunov function method.To facilitate this analysis, we establish the The Lyapunov function L consistently maintains a positive value and only becomes zero at the endemic equilibrium point and differentiating with respect to time t For R oc > 1 , the endemic equilibrium point exists, leading to dL dt is less than zero.It seems that the function L appears as a clear-cut Lyapunov function, suggesting that the endemic equilibrium point reaches asymptotic and global stability.From a biological perspective, this signifies that COVID-19 has remained prevalent in the community over a prolonged duration.

Analysing the sensitivity-only COVID-19 model
We conducted a sensitivity analysis of parameters within the COVID-19 sub-model.The behavior of the model in response to modest changes in a parameter's value is known as the parameter's sensitivity and is tabilize by the symbol φ 1 .It can be expressed as Table 1 displays the data for the sensitivity indices related to the sole COVID-19 sub-model.This sub-model analysis reveals that the COVID-19 contact rate is φ 1 , play a significant role in intensifying the virus's spread.This trend results from an upsurge in secondary infections when these parameters increase, as highlighted by (Martcheva 2015).Conversely, parameters such as τ 1 and µ have a diminishing effect, meaning an uptick in their values could reduce the infection rate.A visual depiction of the sensitivity indices for R oc is showcased in Fig. 2.  Assume S(0) > 0, I k (0) > 0, I k (0) > 0 are positive for time t > 0 and all nonnegative parameters.From the initial condition, all the state variables are nonnegative at the initial time; then, t > 0.
To show the solutions of the model, as it is positive, first, we take dS dt from equation Hence S(0) > 0 , similarly we can prove I k (0) > 0, I k (0) > 0.
Theorem 7 The dynamical system (7) is positively invariant in the closed invariant set.
To obtain an invariant region that shows that the solution is bounded, we have ( 7) www.nature.com/scientificreports/As,t → ∞ , we get 0 ≤ N ≤ � µ , the theory of differential equation 27 in the region.
µ } For the autonomous system representing the Kidney disease-only model, given by ( 7), any solution that starts in Z k will stay within Z k for all t ≥ 0 Kidney disease sub-model with disease-free equilibrium (DFE) By equating Eq. ( 10) to zero dS dt = dI k dt = dI kd dt = 0 The disease-free equilibrium (DFE) of the COVID-19-only model system ( 7) is obtained by setting each of the systems of model system (10) to zero.Also, at the DFE, there are no infections.Thus, the DFE of the COVID-19-only model ( 10) is given by Employing the next-generation matrix method outlined in (Yang 2014), we derive the related next-generation matrix as Consequently, the terms for new infections, F and the subsequent transfer components, V are provided as follows: The next-generation matrix FV −1 's leading eigenvalue, which is also known as the spectral radius, represents the fundamental reproductive number and is defined as: R ok represents the anticipated count of secondary infections produced by a single infected person throughout their entire infectious phase within a wholly susceptible community.

Theorem 8
The DFE is locally asymptotically stable if R Ok < 1 and unstable if R Ok > 1 We use the Jacobian matrix to ascertain the local stability of equilibrium points.For sub-model (7), the Jacobian matrix is given as www.nature.com/scientificreports/At the disease-free equilibrium point 0k , the Jacobian matrix of the sub-model is given For J(� 0k ) the eigenvalues is λ = − μ, and the other eigenvalues can be swiftly obtained using the submatrix We must show that J ′ 2 s trace is negative, and its determinant is positive to determine the local stability of the disease-free equilibrium point. Trc For the kidney disease sub-model, the disease-free equilibrium point is stable when R ok < 1 and unstable when R ok > 1.
Theorem 9 Only when R ok > 1 does the endemic equilibrium point exist?By resolving the above system of equations, we were also able to determine the endemic (disease present) equilibrium point of the renal disease sub-model: µ(µ+σ 1 ) applying this value we get, Hence the endemic equilibrium point exists when R ok > 1

Theorem 10
The disease-free equilibrium point of the Kidney disease-sub model ( 7) is globally asymptotically stable.If R ok < 1 Proof Considering the Lyapunov function Differentiating with respect to time www.nature.com/scientificreports/since all the model parameters are positive, so that dT dt ≤ 0 for R ok ≤ 1 , with dT dt = 0 when I k = I kd = 0 .Using (I k , I kd ) = (0, 0) into the Kidney disease only sub-model (7) represents that S → � µ as t → ∞ .Hence T is a Lyapunov function on 0k and the largest compact invariant set in {(S, I k , I kd ) ∈ � k : dT dt = 0} is 0k .So every solution of (7), with an initial condition in k approaches 0k , as t → ∞ whenever R ok ≤ 1.

Theorem 11
In the kidney disease-only model, the equilibrium point indicating the existence of the disease is globally stable when R 0k > 1.
Denote the endemic equilibrium is denoted by E k = (S * , I * k , I * kd ), At the steady state, the force of infection f k is represented as: In the sub-model (7), we obtain by setting the right-hand sides equal to zero Using (10), The linear Eq. ( 12) has a unique positive solution given by This has biological significance when R ok > 1 .It is mentioned that R ok < 1 implies that φ 2 (θσ 1 + µ) − µ(µ + σ 1 ) < 0. When this occurs, the force of infection f k is negative, suggesting that the dis- ease's equilibrium point shifts to global stability.

Analysis of sensitivity for the kidney disease model
Equation (7) specifies the renal sub-model and the examination of sensitivity for its basic reproduction number uses Yang's (2014) tabilize forward sensitivity index for that basic reproduction number.
Vol.:(0123456789)  2, several observations can be made regarding the factors influencing the spread of kidney disease: 1.The contact rate specific to kidney disease is represented by φ 2 exhibit a pronounced positive correlation with the disease's propagation.This implies that as these rates increase, the disease spreads more aggressively.2. The parameter adjusting for the enhanced transmission of kidney disease among co-infected individuals and those in the end-stage of the disease, denoted as θ , and Progression rates σ 1 also positively influences the spread of the disease.This suggests a higher transfer rate among the co-infected exacerbates the spread of the disease.3. Conversely, certain parameters, namely μ, mitigate the spread of kidney disease.Specifically, elevating the values of this parameter leads to a reduction in the number of individuals afflicted with kidney disease.

COVID-19 and kidney disease full model
By analyzing the equations' right-hand sides, we could derive the equilibrium locations for the entire model (1).
where the forces of infection f k and f c are identical to those in Eqs. ( 5) and (10).The whole model's disease-free equilibrium point (� 0ck ) is then calculated as We have now calculated the basic reproduction number R 0 of the complete model using the next-generation matrix.Using the notation of the diseased states (I c , I k , I kd , I kc , I kdc ) , Given the vector differential equations form is the rate at which new infections arise in compart- ments, V + (x) is the rate at which people are transferred into the compartment, and V − (x) is the rate at which people are transferred out of the compartments I k , I c , I kc , I kd , I kdc , 0, 0, 0, 0, 0, 0) To determine the basic reproduction number R ck of the system, the eigenvalues can be employed, specifi- cally by examining the spectral radius of the matrix FV −1 .The eigenvalues can be determined by assessing the equation: Here eigenvalues are 1 = φ 2 (θ σ 1 +µ) µ(σ 1 +µ) , 2 = φ 1 (τ1+µ) , 3 = 0, 4 = 0, 5 = 0 Thus, it can be concluded that the COVID-19 and kidney disease co-infection model has a reproduction number given by R ck = {R oc , R ok }; where R 0k = φ 2 (θ σ 1 +µ) µ(σ 1 +µ) and R 0c = φ 1 (τ1+µ)

Stability of 0ck for the full co-infection model
Theorem 12 When R ck > 1 , model ( 1) has (� 0ck ) that is locally asymptotically stable.The eigenvalues of each equilibrium were used to examine its local stability (Fudolig and Howard, 2020).The eigenvalues are found in the Jacobian matrix, which each equilibrium has replaced.The model (1)'s Jacobian matrix can be described as www.nature.com/scientificreports/At the disease-free equilibrium, we obtained the following characteristic polynomial: We So, the co-infection full model (1), 0ck reaches local asymptotic stability as a disease-free equilibrium point.

Global stability analysis of co-infection full model
From the full model dX dt = F(X, Z) , dZ dt = T(X, Z), T(X, 0) = 0, Here X = (S, R) and Z = (I k , I c , I kc , I kd , I kdc ) .In this case, representation X, which belongs to R 2 signifies the compartments of healthy individuals, while Z , a part of R 5 , stands for the infected population compartments.The disease-free equilibrium state is denoted by U 0 = (X 0 , 0) , where X 0 = ( � µ , 0) The following assumptions (H 1 ) and (H 2 ) ensure that 0ck for R ck is globally asymptotically stable.(H 1 ) For dX dt = F(X, 0), the equilibrium point U 0 is globally stable; The feasible area of the constructed model is denoted by , and A = D Z T(U 0 , 0) is a Metzler matrix.From our co-infection mathematical model Eq. (1),we From this, condition H 2 is not met.Consequently,U 0 and subse- quently, the disease-free equilibrium point ck , cannot achieve global asymptotic stability.

Parameter estimation
We have derived the values of the model parameters using authentic data from Bangladesh, encompassing both kidney disease and cumulative COVID-19 infected cases.The COVID-19 dataset, from the initial reporting date of March 8, 2020, to September 8, 2020, was collated daily and sourced from 28 .Concurrently, the data for Kidney disease from 2020 to 2023 was compiled every month and can be accessed 29 .To calibrate the model and deduce the parameter values from the data, we employed a hybrid approach combining least squares and Bayesian methods.Additionally, a nonlinear curve-fitting technique was employed, using MATLAB's 'fminsearch' function Certain parameters were inferred from existing literature.For instance, based on Worldometer's data, Bangladesh's average life expectancy in 2020 was 72.72 years (macrotrends,2024), and we considered a subset population of 16,580,000.This led to the calculation of the natural mortality rate per month as the inverse of life expectancy, resulting in a value of µ = 1 72.72×365 = 0.000038 .Furthermore, the recruitment rate was approxi- mated by manipulating the ratio of ∇ µ to yield the initial population, resulting in ∇ = 630 individuals per day.Due to limited data on co-infections, we estimated certain co-infection related parameters, while others were www.nature.com/scientificreports/deduced from actual data.During the estimation process, the initial conditions of the state variables were set as delineated in Table 3. Figure 3 illustrates the model's fit for both cumulative COVID-19 infections and cumulative kidney infections.In Fig. 3a, the model's output for COVID-19 infections is compared to the actual observed data of COVID-19 cases.Similarly, Fig. 3b demonstrates the alignment between the model's simulation and the observed data for kidney disease infections.In both instances, solid lines represent the model's simulated output, and dotted lines correspond to the actual observed data for the two diseases from Bangladesh.The comparison reveals a strong congruence between the model simulations and the actual data.

Numerical simulations
To explore the co-infection dynamics between COVID-19 and Kidney disease in scenarios without treatment, we carried out numerical simulations using the combined COVID-19 and Kidney disease model.The majority of theoretical results from this investigation are illustrated through these simulations.For our computational study, we employed the ode45 function.Ode45, incorporated into MATLAB, is a non-stiff one-step solver based on the Runge-Kutta (4, 5) method.It stands out for its speed, accuracy, and stability.While it is superior to the Euler method in terms of efficiency, its true strength lies in its simplicity and stability, especially when juxtaposed with multi-step strategies.Despite consuming more computational time than other equivalent accuracy multi-step methods, the straightforward nature and user-friendliness of ode45 compensate for its computational demands.Parameters driving our simulations can be found in Table 3, along with the initial conditions set for the experiment S = 50000, I c = 500, I k = 300, I kd = 200, I kdc = 100, R = 30.
Figure 4 showcases a series of time-dependent plots that illustrate the dynamics of the co-infection as it evolves over time.These plots have been constructed by numerically solving the co-infection model represented by Eq. (1).The solutions have been derived using the specific parameter values enumerated in Table 3.The progression depicted in each plot provides insights into the tabiliz of the diseases in the system and their interactions over the duration captured.Figures 5, 6 and 7 illustrate the stability characteristics solutions when subject to many initial circumstances.Specifically, Fig. 5 focuses on the initial conditions for the susceptible compartment within the COVID-19 sub-model.In contrast, Fig. 6 pertains to the infected compartment of the same sub-model.Lastly, Fig. 7 delves into the dynamics of those co-infected with COVID-19 and the primary stage of kidney disease.These figures provide valuable insights into how the system responds to changes in initial states, shedding light on the disease dynamics and potential interactions between the two health conditions.
Figures 8 and 9 elucidate the influence of rates φ 1 and φ 2 on co-infected individuals within the I kc class.Nota- bly, as these rates escalate, initially there's a consequent increase in the count of individuals in the co-infected population, later after reaching a peak the count of individuals gradually declines.These rates, presumably, describe how quickly individuals leave or transition out of this co-infected population.The main observation drawn from Fig. 8 is that, as the φ 1 increases, the rate of increase in the number of co-infected individuals also increases sharply and reaches a peak at almost the same time, declining gradually as the infected individuals recover or die.In context, the effect of φ 2 on the number of infected individuals is very sensitive.As is noted in Fig. 9, for the largest value of φ 2 the number of infected individuals quickly arrives at the peak.As φ 2 decreases, it takes a relatively greater time for the number of co-infected individuals to reach at peak.  3.As φ 1 increases, the co-infected individuals also increase for a certain period (around 130 days) and then decrease slowly.On the contrary, the dynamics of co-infected individuals in the end-stage kidney disease show some variation.For example, the largest value of φ 2 there is an increasing trend in the number of co-infected individuals, so for moderate value of φ 2 .But for the lowest value of φ 2 the trend of co-infected individuals shows up and down tabiliz.Interestingly, contrary to initial assumptions, the figures indicate that a rise in either φ 1 or φ 2 corresponds to an uptick in the number of co-infected individuals within the I kdc class.
In Fig. 12, we illustrate the influence of transfer rates to the co-infected class, stemming from each actively infected individual of the respective diseases.Specifically, this figure delves into the effects of contact rates about co-infected compartments of both COVID-19 and end-stage kidney disease ( I kdc ) as described in our system (1).The interactive effect of φ 1 and φ 2 depicts that the co-infected population rises to a peak at a particular time point and then decreases regardless of the different parameter combinations.However, the curves do not intersect for different levels of either φ 1 or φ 2 the trend of co-infected populations is similar for I kdc group.
Figure 13 showcases the proliferation of co-infected as the effective contact rates vary.In contrast, the dynamics of individuals solely infected with COVID-19, to differing contact rates, are depicted in both Figs. 12 and 13.A notable observation is that the population in the state I c diminishes while in I kdc grows as transmission coef- ficients escalate.Crucially, these numerical observations echo our analytical insights drawn from the sensitivity analysis within the sub-models.Figure 14 shows an inverse relationship between the susceptible and COVID-19-infected populations.As the number of susceptible individuals rises, the number of those infected with COVID-19 decreases.This phase plane suggests that new infections decline as more individuals become less vulnerable or exposed to the virus (perhaps due to factors like vaccination, prior infection, or preventive measures).
In Fig. 15, the graph reveals that as the number of infected solely with only COVID-19 grows, there is a corresponding increase in the population co-infected with both COVID-19 and the primary stage of kidney disease.Simultaneously, we observe a decline in the susceptible population.Intriguingly, when there's a surge in the susceptible population, neither the co-infected nor the solely COVID-19-infected group shows a proportional rise.Instead, their numbers stabilise or remain consistent; they plateau or stay steady.
In Fig. 16, the scatter plot shows a positive correlation between the number of kidney disease individuals and the number of COVID-19-infected people.Also, our analytical analysis shows that people with kidney disease are more likely to get COVID-19.
Figure 17 demonstrates a correlation between the two variables, suggesting that those infected with COVID-19 have a higher risk of being co-infected with COVID-19 and kidney disease.COVID-19 can damage the kidneys, leading to acute kidney injury and a sudden loss of kidney function.Acute kidney injury can be fatal and is more likely to occur in people with kidney disease.3.  3.

Conclusion
We developed a mathematical model to study the spread of co-infection between kidney disease and COVID-19.This model ensures solutions are positive and limited within a biologically meaningful range.We identified equilibrium points for the diseases separately and analysed their stability based on their basic reproduction numbers.We also examined the co-infection reproduction number and its sensitivity analysis, revealing that a rise in infection rates from either disease increases the co-infection risk.The key findings of the new development model are listed below; • Our analysis found that if the infection rate for either COVID-19 or kidney disease increases, the risk of people getting both diseases increases significantly.This means that controlling the spread of each disease is crucial to reducing the overall risk of co-infections.• Changing how easily each disease is transmitted (known as transmission coefficients) affects the diseases differently depending on their stage.For example, a transmission change might significantly impact someone who's just contracted the disease more than someone living with it for a while.• We looked at how changes in the contact rate for COVID-19 (φ 1 ) and the contact rate for kidney disease (φ 2 ) affect the diseases.We found that these changes have different impacts depending on whether the kidney disease is in its early stage (primary) or late stage (end-stage).This means how each disease spreads and affects people can vary significantly based on the disease's progression.• We identified specific points, called equilibrium points, for each disease.These points help us understand how likely the disease will remain in the population over time.If the number we calculate for these points is more than one, it suggests that the disease will continue to exist within the population.This is a key indicator for public health strategies, highlighting the need for ongoing disease management and control measures.
The observations drawn from the model are consistent with analytical conclusions from the sensitivity analysis, especially emphasising the critical role of reducing the susceptible population-potentially through measures like vaccination or natural immunity-to decrease new infections.The findings highlight the complex interplay of disease transmission and co-infections, presenting areas of concern and possible intervention points for effective disease control.The present results and models also maximise the benefits of simulation modelling to minimise the global health complexity of COVID-19 and kidney disease.The more effective strategies for reducing the impact of COVID-19 and kidney disease through optimal control methods are used in our forthcoming studies.

0 ∆Figure 1 .Theorem 2
Figure 1.Flow chart for the transmission dynamics of the co-infection of COVID-19 with kidney disease.

Theorem 6
www.nature.com/scientificreports/Kidneydisease-only modelKidney disease-only sub-model from the co-infection model, we get I c = 0, I kc = 0, I kdc = 0, R = 0 All the populations of the system with positive initial conditions are nonnegative.

Figure 2 .
Figure 2. The graphical depiction of the sensitivity indices concerning the primary reproduction number (R oc ) parameters are shown in the COVID-19 sub-model.

Figure 3 .
Figure 3. Model fitting with reported COVID-19 and kidney disease data.

Figure 4 .
Figure 4. Solution of the comprehensive co-infection model using parameter values in Table3.

Figure 5 .
Figure 5. Graphical representation of the stability at the disease-free equilibrium point when R ck < 1 and R ck > 1.

Figures 10
Figures 10 and offer detailed visual representations of how the rates φ 1 and φ 2 affect the co-infected indi- viduals within the COVID-19 and end stage of kidney disease (I kdc ) class.These rates signify how individuals transition out of the co-infected state.As φ 1 increases, the co-infected individuals also increase for a certain period (around 130 days) and then decrease slowly.On the contrary, the dynamics of co-infected individuals in the end-stage kidney disease show some variation.For example, the largest value of φ 2 there is an increasing trend in the number of co-infected individuals, so for moderate value of φ 2 .But for the lowest value of φ 2 the trend of co-infected individuals shows up and down tabiliz.Interestingly, contrary to initial assumptions, the figures indicate that a rise in either φ 1 or φ 2 corresponds to an uptick in the number of co-infected individuals within the I kdc class.In Fig.12, we illustrate the influence of transfer rates to the co-infected class, stemming from each actively infected individual of the respective diseases.Specifically, this figure delves into the effects of contact rates about co-infected compartments of both COVID-19 and end-stage kidney disease ( I kdc ) as described in our system (1).The interactive effect of φ 1 and φ 2 depicts that the co-infected population rises to a peak at a particular time point and then decreases regardless of the different parameter combinations.However, the curves do not intersect for different levels of either φ 1 or φ 2 the trend of co-infected populations is similar for I kdc group.Figure13showcases the proliferation of co-infected as the effective contact rates vary.In contrast, the dynamics of individuals solely infected with COVID-19, to differing contact rates, are depicted in both Figs. 12 and 13.A notable observation is that the population in the state I c diminishes while in I kdc grows as transmission coef- ficients escalate.Crucially, these numerical observations echo our analytical insights drawn from the sensitivity analysis within the sub-models.

Figure 6 .
Figure 6.Graphical representation of the stability at the disease-free equilibrium point when R 0c < 1 and R 0c > 1.

Figure 7 .
Figure 7. Graphical representation of the stability at the disease-free equilibrium point when R ck < 1 and R ck > 1.

Figure 8 .
Figure 8. Behavior of I kc for the different values of φ 1 and other values of the parameter in Table3.

Figure 9 .Figure 10 .
Figure 9. Behavior of I kc for the different values of φ 2 and other parameter values in Table3.

Figure 11 .
Figure 11.Effect of Contact rate of kidney disease interventions on co-infected populations.

Figure 12 .
Figure12.Impact of the contact rates φ 1 and φ 2 on the transmission dynamics of the co-infected ones ( I kdc )

Figure 13 .
Figure 13.Impacts of the contact rates φ 1 and β 2 on the dynamics of infected COVID-19 (I c ) transmission.

Figure 14 .
Figure 14.Phase plane illustrating the dynamical interplay between susceptible population individuals S and infected COVID-19 individuals I c .

Figure 15 .
Figure 15.Phase portrait illustrating the dynamic interactions among the compartments S, I c and I kc .

Figure 16 .
Figure 16.Phase portrait illustrating the dynamic interactions among the compartments I k and I c .

Figure 17 .
Figure 17.Phase plot illustrating the dynamic interactions among the compartments I c and I kdc . 1) Table 3 were used to compute the sensitivity indices for the only COVID-19 submodel.

Table 2 .
Sensitivity indices for the kidney disease-only sub-model.

Table 3 .
Description of variables and parameters in the model equation.